It seems you are taking back your proposal of how storage works, so we are now back to what do you mean by stored, and what is doing the storing?
I don't like the word storing, because I think of it more as reading a physical system state and mapping to a thought. Don't be confused by the infinity of potential physical states in a finite space. Instead, how can these states be interpreted and mapped? Let's take a tiny sphere, square, or some other space and call it a thought machine, with input a physical state and output to a thought. How do we read the physical input?
We need to break the space into smaller units so we can register difference in the space. These are the sensors, which have the properties of measuring physical space, and their sum readings output a thought. Subdivisions of the space, and the subdivisions having properties contributing to the whole, can't be avoided if the object having the thoughts is the same. Why fire a thought at one point, based on one physical state, or why fire another, based on another state?
We need to avoid the infinite regress, so these sensors don't have their own sensors, and are the smallest unit by which the thought machine can interpret space. From these indivisible units of space, however many there are, we can then build our thought machine. This is what I mean when I say from the "bottom up". Start with the sensors and then figure out the machine. Don't take the space the thought machine operates in and keep subdividing (from the "top down") to accommodate any state you wish.
For what I think is frequently missed is that the thought machine has its own properties independent of what is actually in a physical space. First, its sensors don't follow around the objects being measured. We don't redraw the sensors as the chess piece moves, for example. If the chess piece's sensor was in the same place relative to the chess piece, how would it register change?
Also, the machine is always firing and always measuring a state, independent of what is physically there. Take a square in space divided into two equal halves, and that same square also divided into three equal thirds. Let's say enough matter moves into the square so that half is filled. Sensors measuring matter for the halves would work, but the ones for the thirds would not, as we have to account for at least one sensor for one third needing knowledge of smaller intervals. We can't switch the sensors for the third back on if we later have enough matter for a 1/3 or 2/3 of the square, as that means the sensors for the halves would not work. We need to pick one fractional division of the square and stick with it.
Thus, in short, two axioms about sensors are:
1) They are the lowest interval by which a thought machine can measure and interpret distance. They can't process any states that require knowledge of smaller intervals.
2) They always give a reading, and their properties are independent of the item being measured.